Invertibility Properties of Singular Integral Operators Associated with the Lamé and Stokes Systems on Infinite Sectors in Two Dimensions

Irina Mitrea, Katharine Ott, Warwick Tucker

Research output: Contribution to journalArticleResearchpeer-review

Abstract

In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on Lp(∂Ω) , 1 < p< ∞, whenever Ω is an infinite sector in R2. This analysis is relevant to the layer potential treatment of a variety of boundary value problems for the Lamé system of elastostatics and the Stokes system of hydrostatics in the class of curvilinear polygons in two dimensions, such as the Dirichlet, the Neumann, and the Regularity problems. Mellin transform techniques are used to identify the critical integrability indices for which invertibility of these layer potentials fails. Computer-aided proofs are produced to further study the monotonicity properties of these indices relative to parameters determined by the aperture of the sector Ω and the differential operator in question.

Original languageEnglish
Pages (from-to)151-207
Number of pages57
JournalIntegral Equations and Operator Theory
Volume89
Issue number2
DOIs
Publication statusPublished - 1 Oct 2017
Externally publishedYes

Keywords

  • Computer-aided proof
  • Conormal derivative
  • Double layer potential
  • Hardy kernel operator
  • Infinite sector
  • Interval analysis
  • Lamé system
  • Mellin transform
  • Pseudo-stress conormal derivative
  • Single layer potential
  • Stokes system
  • Validated numerics

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